(1) Field of the Invention
The present invention relates to a method of constructing elements which wipe each other during corotation about two parallel axes in such a manner that they constantly touch each other at at least one point.
(2) Description of Related Art
In the case of two circles which are arranged adjacent to each other on two parallel axes, as depicted diagrammatically in FIG. 1, it is generally known that such circles wipe each other during corotation in such a manner that they constantly touch each other at a point located between the centres of rotation of the circles.
It is also known that not only circles but also a number of additional geometrical configurations exist which constantly touch each other at one point during their corotation. One example of such configurations is shown in FIG. 2.
The two-dimensional geometrical configurations shown in FIGS. 1 and 2 can be extended in the third dimension in different ways. One simple alternative is, for example, to extend the configurations linearly in the direction of the axes of rotation to form disc-like elements which wipe each other during corotation along a line between the centres of rotation which runs parallel to the axes of rotation.
Another alternative is, for example, to extend the geometrical configurations in a screw-like fashion along the axes of rotation, so that screw-like elements are formed which touch each other along a curve between the elements.
Such elements which constantly touch each other at at least one point during their corotation about parallel axes are of particular importance in extruder technology where they are used as screw extruders, for example, for processing viscous materials or for mixing purposes. Such corotating twin- and multi-screw extruders are known to those skilled in the art from patent and technical literature. The following publication [1] can be mentioned as an example in this connection: K. Kohlgrüber: “Der gleichläufige Doppelschneckenextruder” (“Corotating twin-screw extruders”), Publishers: Hanser Verlag, 2007. In screw extruders, the property that adjacent screws wipe each other in pairs during corotation has the advantage that they reciprocally scrape and therefore clean each other.
Rules exist for the construction of selected elements which constantly touch each other at at least one point during their corotation about parallel axes.
Thus, it is for example known from the literature on screw extruders (see for example pages 96 to 98 of [1]) that a screw element with a cross-sectional profile of the geometrical configuration of FIG. 2 of the present application can be constructed from arcs.
The general criteria which have to be fulfilled in order to ensure that two elements corotating about two parallel axes constantly touch each other at at least one point have however not so far been disclosed.
It is known (see for example [2]: Booy “Geometry of fully wiped twin-screw equipment”, Polymer Engineering and Science 18 (1978) 12, pages 973-984) that the corotation of two intercontacting elements about their stationary axes is kinematically equivalent to the “movement without rotation” of the one element about the other—in this case stationary—element. This special phenomenon can be used for the stepwise generation of geometrical configurations which constantly touch each other at one point during their corotation. During this process the first (“generated”) configuration remains stationary and the second (“generating”) configuration is translationally moved about the first configuration on an arc. It is possible to predefine part of the profile of the second configuration and determine which profile is thereby generated on the first configuration. The generated configuration is, as it were, “carved” out by the generating configuration.
No general method as to how to generate the actual predefined part of the second configuration has, however, so far been disclosed. Although [2] describes one possible approach for generating the starting profile section from which the remaining profile is generated, this approach is mathematically highly laborious and above all not universally applicable, i.e. it is only possible to generate such profiles capable of being defined by the mathematical functions specified in [2].
Based on the prior art, the problem therefore arose of providing a general method of constructing elements which constantly touch each other at at least one point during their corotation.
Surprisingly, the basic principles underlying such elements which constantly touch each other at at least one point as they corotate about parallel axes have now been found. These basic principles make it possible to formulate a simple method of constructing such elements. Surprisingly it has been found that such elements can be constructed simply on paper using a pair of compasses and an angle ruler, without the need for complicated calculations.